Improper dihedral angles: how to tell between 0 and 180 degress

How can I tell the difference between a 0 and 180 degree value when looking at an improper dihedral. For example, in the molecule below would the dihedral angle 4-2-3-1 have a value of 0 or 180 degrees?

It is obvious that the two planes defined by 4-2-3 and 2-3-1 are coplanar, but how do I define the angle between them? If I understand correctly, it should be the angle between the norm vectors of the two planes, but I am not sure how to choose the direction of the norms. Is there a standard convention for this?

1 Answer

Yes. Think of the hinge 2-3 as dividing the plane into two half-planes. If 1 and 4 are in the same half-plane as if the atoms had been folded over, that is a zero dihedral angle. If 2 and 4 are in opposite half-planes like an unfolded sheet, that is a 180° dihedral angle. Your drawing looks like the latter.

• Yes, following that through, you can define a dihedral 1 3 4 2 alternatively, being 0°. That doesn't make much sense in the case of ethyne, the model shown here, but in BH3 the definition would make more sense. Sep 12, 2022 at 18:10
• I'm quite sure there is a mathematical way of showing this, but that is just a bit too much for me to formalize. Essentially when you go clockwise, the normal vector sticks out of the plane to the front, counter clockwise to the back. Then it's easy to form an angle. Sep 12, 2022 at 18:15
• Exactly @Martin. Given four vertices of a tetrahedron you can define any of six distinct dihedral angles! Sep 12, 2022 at 18:15
• @OscarLanzi how about 180 and -180 degree? Are they same? Jul 11, 2023 at 3:33